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ReLU Networks for Model Predictive Control: Network Complexity and Performance Guarantees

Xingchen Li, Keyou You

TL;DR

This work tackles the problem of certifying ReLU neural networks as approximations of MPC policies for constrained LTI systems by deriving explicit width and depth bounds that guarantee feasibility and closed-loop stability under a uniform error bound. It introduces a projection-based constraint-tightening mechanism and a state-dependent Lipschitz property for the MPC cost to enable sharper convergence analysis, achieving an invariant set size of $\mathcal{O}(\delta_1)$. To further reduce network complexity while preserving performance, the authors propose a non-uniform, state-aware error framework with input/output scaling that concentrates approximation resources near the origin and yields smaller invariant sets with fewer network resources. The results are complemented by a numerical example illustrating depth versus width trade-offs and the benefits of state-aware scaling, supporting the practical viability of certifiable ReLU-NN MPC in real-time control settings.

Abstract

Recent years have witnessed a resurgence in using ReLU neural networks (NNs) to represent model predictive control (MPC) policies. However, determining the required network complexity to ensure closed-loop performance remains a fundamental open problem. This involves a critical precision-complexity trade-off: undersized networks may fail to capture the MPC policy, while oversized ones may outweigh the benefits of ReLU network approximation. In this work, we propose a projection-based method to enforce hard constraints and establish a state-dependent Lipschitz continuity property for the optimal MPC cost function, which enables sharp convergence analysis of the closed-loop system. For the first time, we derive explicit bounds on ReLU network width and depth for approximating MPC policies with guaranteed closed-loop performance. To further reduce network complexity and enhance closed-loop performance, we propose a non-uniform error framework with a state-aware scaling function to adaptively adjust both the input and output of the ReLU network. Our contributions provide a foundational step toward certifiable ReLU NN-based MPC.

ReLU Networks for Model Predictive Control: Network Complexity and Performance Guarantees

TL;DR

This work tackles the problem of certifying ReLU neural networks as approximations of MPC policies for constrained LTI systems by deriving explicit width and depth bounds that guarantee feasibility and closed-loop stability under a uniform error bound. It introduces a projection-based constraint-tightening mechanism and a state-dependent Lipschitz property for the MPC cost to enable sharper convergence analysis, achieving an invariant set size of . To further reduce network complexity while preserving performance, the authors propose a non-uniform, state-aware error framework with input/output scaling that concentrates approximation resources near the origin and yields smaller invariant sets with fewer network resources. The results are complemented by a numerical example illustrating depth versus width trade-offs and the benefits of state-aware scaling, supporting the practical viability of certifiable ReLU-NN MPC in real-time control settings.

Abstract

Recent years have witnessed a resurgence in using ReLU neural networks (NNs) to represent model predictive control (MPC) policies. However, determining the required network complexity to ensure closed-loop performance remains a fundamental open problem. This involves a critical precision-complexity trade-off: undersized networks may fail to capture the MPC policy, while oversized ones may outweigh the benefits of ReLU network approximation. In this work, we propose a projection-based method to enforce hard constraints and establish a state-dependent Lipschitz continuity property for the optimal MPC cost function, which enables sharp convergence analysis of the closed-loop system. For the first time, we derive explicit bounds on ReLU network width and depth for approximating MPC policies with guaranteed closed-loop performance. To further reduce network complexity and enhance closed-loop performance, we propose a non-uniform error framework with a state-aware scaling function to adaptively adjust both the input and output of the ReLU network. Our contributions provide a foundational step toward certifiable ReLU NN-based MPC.
Paper Structure (33 sections, 14 theorems, 82 equations, 10 figures, 1 table)

This paper contains 33 sections, 14 theorems, 82 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. MPC for quadratic stabilization of constrained LTI systems
  4. The ReLU NN-based MPC policy
  5. Objective of this work
  6. Network Complexity for the ReLU NN-based MPC Policy in Solving Problem \ref{['problem1']}
  7. Constraint satisfaction via set tightening and projection
  8. State-dependent Lipschitz continuity of $v_\text{mpc}(\cdot)$ for improved closed-loop convergence
  9. Explicit bounds on the required ReLU network complexity
  10. Network Complexity for the ReLU NN-based MPC Policy in Solving Problem \ref{['problem2']}
  11. The motivation of using non-uniform error bound
  12. State-aware scaling for non-uniform error bounds
  13. Scaling back the output of ReLU network
  14. Scaling the input of ReLU network
  15. An illustrative example
  16. ...and 18 more sections

Key Result

Lemma 1

Let $\epsilon \le \delta_1 d(\mathcal{U})/(2D(\mathcal{U}))$. If then $\| u_{\text{nn}}(x) - u_{\text{mpc}}(x) \| \leq \delta_1$ and $u_{\text{nn}}(x) \in \mathcal{U}$, i.e., equ:con2 and equ:con1 hold.

Figures (10)

  • Figure 1: The MPC and the ReLU NN-based MPC
  • Figure 2: Illustration of the main challenges and our analytical approach.
  • Figure 3: Feasible combinations of $n_w'$ and $n_d'$ satisfy \ref{['equ:main']}.
  • Figure 4: The non-uniform scaling approach (The definitions of $\widetilde{u}_\text{mpc}(\cdot)$ and $\widetilde{u}_\text{nn}(\cdot)$ are given in \ref{['equ:trans_recover_first']} and \ref{['equ:transformation']}, respectively.)
  • Figure 5: An example of $\beta(x)=\delta_2(x)/\bar{\delta}$ and the input scaling function $T(x)$.
  • ...and 5 more figures

Theorems & Definitions (33)

  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2: State-dependent Lipschitz continuity of $v_\text{mpc}$
  • proof
  • Lemma 3
  • proof
  • Theorem 1: Network Complexity for Problem \ref{['problem1']}
  • proof
  • Lemma 4
  • ...and 23 more